Theorem ntrclscls00 44740

Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that the closure of the empty set is the empty set hold equally. (Contributed by RP, 1-Jun-2021.)

Hypotheses

Ref	Expression
ntrcls.o	⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))
ntrcls.d	⊢ 𝐷 = (𝑂‘𝐵)
ntrcls.r	⊢ (𝜑 → 𝐼𝐷𝐾)

Assertion

Ref	Expression
ntrclscls00	⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ (𝐾‘∅) = ∅))

Distinct variable groups:   𝐵,𝑖,𝑗,𝑘   𝑗,𝐼,𝑘   𝑗,𝐾,𝑘   𝜑,𝑖,𝑗,𝑘

Allowed substitution hints:   𝐷(𝑖,𝑗,𝑘)   𝐼(𝑖)   𝐾(𝑖)   𝑂(𝑖,𝑗,𝑘)

Proof of Theorem ntrclscls00

Step	Hyp	Ref	Expression
1		ntrcls.o	. . . . . 6 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))
2		ntrcls.d	. . . . . 6 ⊢ 𝐷 = (𝑂‘𝐵)
3		ntrcls.r	. . . . . 6 ⊢ (𝜑 → 𝐼𝐷𝐾)
4	1, 2, 3	ntrclsfv1 44729	. . . . 5 ⊢ (𝜑 → (𝐷‘𝐼) = 𝐾)
5	4	fveq1d 6883	. . . 4 ⊢ (𝜑 → ((𝐷‘𝐼)‘∅) = (𝐾‘∅))
6	2, 3	ntrclsbex 44708	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
7	1, 2, 3	ntrclsiex 44727	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
8		eqid 2761	. . . . 5 ⊢ (𝐷‘𝐼) = (𝐷‘𝐼)
9		0elpw 5326	. . . . . 6 ⊢ ∅ ∈ 𝒫 𝐵
10	9	a1i 11	. . . . 5 ⊢ (𝜑 → ∅ ∈ 𝒫 𝐵)
11		eqid 2761	. . . . 5 ⊢ ((𝐷‘𝐼)‘∅) = ((𝐷‘𝐼)‘∅)
12	1, 2, 6, 7, 8, 10, 11	dssmapfv3d 44693	. . . 4 ⊢ (𝜑 → ((𝐷‘𝐼)‘∅) = (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))))
13	5, 12	eqtr3d 2798	. . 3 ⊢ (𝜑 → (𝐾‘∅) = (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))))
14		dif0 4333	. . . . . . 7 ⊢ (𝐵 ∖ ∅) = 𝐵
15	14	fveq2i 6884	. . . . . 6 ⊢ (𝐼‘(𝐵 ∖ ∅)) = (𝐼‘𝐵)
16		id 23	. . . . . 6 ⊢ ((𝐼‘𝐵) = 𝐵 → (𝐼‘𝐵) = 𝐵)
17	15, 16	eqtrid 2808	. . . . 5 ⊢ ((𝐼‘𝐵) = 𝐵 → (𝐼‘(𝐵 ∖ ∅)) = 𝐵)
18	17	difeq2d 4080	. . . 4 ⊢ ((𝐼‘𝐵) = 𝐵 → (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))) = (𝐵 ∖ 𝐵))
19		difid 4331	. . . 4 ⊢ (𝐵 ∖ 𝐵) = ∅
20	18, 19	eqtrdi 2812	. . 3 ⊢ ((𝐼‘𝐵) = 𝐵 → (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))) = ∅)
21	13, 20	sylan9eq 2816	. 2 ⊢ ((𝜑 ∧ (𝐼‘𝐵) = 𝐵) → (𝐾‘∅) = ∅)
22		pwidg 4581	. . . . 5 ⊢ (𝐵 ∈ V → 𝐵 ∈ 𝒫 𝐵)
23	6, 22	syl 18	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐵)
24	1, 2, 3, 23	ntrclsfv 44733	. . 3 ⊢ (𝜑 → (𝐼‘𝐵) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝐵))))
25	19	fveq2i 6884	. . . . . 6 ⊢ (𝐾‘(𝐵 ∖ 𝐵)) = (𝐾‘∅)
26		id 23	. . . . . 6 ⊢ ((𝐾‘∅) = ∅ → (𝐾‘∅) = ∅)
27	25, 26	eqtrid 2808	. . . . 5 ⊢ ((𝐾‘∅) = ∅ → (𝐾‘(𝐵 ∖ 𝐵)) = ∅)
28	27	difeq2d 4080	. . . 4 ⊢ ((𝐾‘∅) = ∅ → (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝐵))) = (𝐵 ∖ ∅))
29	28, 14	eqtrdi 2812	. . 3 ⊢ ((𝐾‘∅) = ∅ → (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝐵))) = 𝐵)
30	24, 29	sylan9eq 2816	. 2 ⊢ ((𝜑 ∧ (𝐾‘∅) = ∅) → (𝐼‘𝐵) = 𝐵)
31	21, 30	impbida 812	1 ⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ (𝐾‘∅) = ∅))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1568   ∈ wcel 2141  Vcvv 3453   ∖ cdif 3901  ∅c0 4285  𝒫 cpw 4561   class class class wbr 5108   ↦ cmpt 5191  ‘cfv 6536  (class class class)co 7410   ↑_m cmap 8823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7985  df-2nd 7986  df-map 8825

This theorem is referenced by: (None)